Abstract
This letter introduces a new efficient algorithm for the two-dimensional weighted Laguerre polynomials finite difference time-domain (WLP-FDTD) method based on domain decomposition scheme. By using the domain decomposition finite difference technique, the whole computational domain is decomposed into several subdomains. The conventional WLP-FDTD and the efficient WLP-FDTD methods are, respectively, used to eliminate the splitting error and speed up the calculation in different subdomains. A joint calculation scheme is presented to reduce the amount of calculation. Through our work, the iteration is not essential to obtain the accurate results. Numerical example indicates that the efficiency and accuracy are improved compared with the efficient WLP-FDTD method.
1. Introduction
The Laguerre-based finite difference time-domain (WLP-FDTD) method [1] is widely used in solving electromagnetic problems with fine structures. As it is not constrained by the Courant-Friedrich-Levy condition, the WLP-FDTD method is unconditionally stable.
However, the WLP-FDTD method often produces a huge sparse matrix equation in the process of solving the fine structure problem. Directly solving the associate matrix is a challenging work. In order to settle this problem, He et al. introduced the domain decomposition finite difference technique to the WLP-FDTD method [2]. The huge sparse matrix is decomposed into several small ones. Duan et al. investigated an efficient WLP-FDTD method [3], in which the huge sparse matrix is turned into tridiagonal matrixes by introducing a perturbation term. CPU time and memory storage can be saved greatly. However, the splitting error caused by the perturbation term turns to be pronounced in the fine structure area. To improve this efficient method, Chen et al. studied a new splitting scheme by introducing a new perturbation term [4]. However, the iteration is required in this method, which increases the computational time.
In this letter, we introduce a new efficient algorithm for the 2D WLP-FDTD method. The whole computational domain is decomposed into several subdomains utilizing the domain decomposition scheme. In order to eliminate the splitting error, the conventional WLP-FDTD method [1] is applied to the subdomains where the fields have larger spatial derivatives. And to speed up the calculation, the efficient WLP-FDTD method [4] is used to the other subdomains where the splitting error is limited. Meanwhile, we devise a joint calculation scheme to further reduce the computational time. Through using this approach, the iteration is no longer necessary to get the accurate results. Numerical example shows that the accuracy can be improved and the CPU time is reduced to about 52.7% of the original one [4].
2. Theories
2.1. Domain Decomposition Technique
The domain decomposition technique is similar to that in [2]; the computational domain is decomposed into nine subdomains, as shown in Figure 1. However, in this paper, the conventional WLP-FDTD method is applied to (fine structure exists). Meanwhile, the efficient WLP-FDTD method is used to the other subdomains and the interface , where .
In the simple and lossless medium, Maxwell’s equations of 2D wave for the Laguerre-FDTD can be deduced as [4]where is the free space electric permittivity, is the free space magnetic permeability, and is the electric excitation source. The time-scale factor and order of the Laguerre polynomials are defined as and , respectively.
Rewriting (1a) and (1b) as the matrix equations, we havewhere vector and represent the unknown field variables and , respectively.
The conventional WLP-FDTD equations [1] can be written aswhere vector represents the unknown field variables and .
According to (1a), in the subdomains and , in (2a) correspond to the interior to interior coupling. So firstly, in these subdomains can be solved by (2a) independently. Therefore one can have the sparse matrix system aswhere , and , respectively, represent the field vector in and , in and , and in . and represent the matrix in (2a) corresponding to the interior to interior coupling in and . and represent the matrix in (2b) corresponding to the interior to interior coupling in and . represents the matrix in (3) corresponding to the interior to interior coupling in . According to the efficient WLP-FDTD method [4], , , , and are tridiagonal matrixes. The global assembly of the interior-interface and interface-interior coupling is defined as and , respectively [5]. corresponds to the interface-interface coupling for .
The efficient WLP-FDTD equation (1a) is applied to the interfaces and ; therefore the coupling terms for the vector (- the interior of and - the interior of ) do not exist in the sparse matrix system. However, the equations for vector in and have the coupling term of the interior-interface . According to (1b), the equation for vector in and can be written as where and are the vector in and . ; . and are the vector in and . and correspond to the interior of - and the interior of - coupling.
According to (1a), the equations for vector in and can be written as where and .
For (4), the Schur complement system is given by [6]where
Once (7) has been solved, (4) can be solved by Then (5a) and (5b) can be solved. After that, all the calculations of and are completed.
2.2. Discussion about Reducing the Amount of Calculation
(5a), (5b), (6a), (6b), (9a), and (9b) are tridiagonal equations, so in this paper, the new algorithm has greatly decreased the amount of calculation compared with [2]. However, the matrix inversion is required in (8a) and (8b), and it is still very time consuming. In order to avoid solving the inverse matrix directly and further cut down the unnecessary calculation, we introduce a joint calculation scheme of (8a), (8b), (9a), (9b), and (9c). In this section, we take , , and (9a), for example.
The matrix can be written aswhere and . is a tridiagonal matrix which can be decomposed as follows:
According to (1a), and have only nonzero elements. Through some complex manipulations, one can obtain
where and represent the nonzero elements in and corresponding to , respectively. (10)~(12b) indicate that the calculation of is quite simple and time saving due to the efficient WLP-FDTD method in .
Based on (10) in , (9a) can be written as
Only one nonzero element is contained in ; one can have
Set to meet the following equation:then one can obtain obviously
As there is only one nonzero element in corresponding to , one can havewhere is the last element of in (15).
Equations (13) and (15) are tridiagonal equations and the coefficient matrixes are all . According to (13)~(15), only the last element is different between and
Therefore, we use the chasing method to the joint calculation of (13) and (15). The procedures are as follows:
It can be seen from (19a)~(19c) that the calculation of is independent of . By inserting the results into (17), one can obtain . The calculation of is dependent on ; therefore once (7) has been solved, (19d)~(19f) can be solved. The method we introduced above avoids solving directly. And due to the joint calculation scheme, in addition to (12a), (12b), (17), (19b), and (19c), the total amount of calculation of , , and (13) is the same as solving (13) alone. Therefore, the method has cut down the unnecessary calculation effectively. In the same way, one can deal with the other parts of , , (9a), and (9b).
3. Numerical Results
In order to validate the effectiveness of the new algorithm, a numerical example is implemented. A parallel plate waveguide with perfect electric conductor (PEC) slot of 0.2 mm is shown in Figure 2, the same as [4]. The PEC is infinitely thin. In order to simulate this fine structure, we take the graded mesh with an expansion factor for the -axis ( for the -axis). In the and directions, there are 250 and 130 subdivisions including uniform cells for the PEC slot, and there are 10 and 40 subdivisions in . The size of the minimum grid is 5 mm × 10 m. Mur’s first-order absorbing boundary conditions [7] are set to end the boundaries. The excitation is located at the position of and the wave is sinusoidal modulated Gaussian pulse:where , , and GHz. The time duration ns, , and [8, 9] are chosen for the conventional WLP-FDTD and the proposed method, and ps is set to solve the Laguerre coefficient of the excitation pulse.
Figure 3 shows the vector of the electric field at the observation point. Good agreement can be found between the conventional WLP-FDTD, the efficient WLP-FDTD [4], and the proposed method.
Table 1 illustrates the CPU time for the numerical simulations. It can be observed that the CPU time for the proposed method is reduced to about 15.9% of the conventional WLP-FDTD and 52.7% of the efficient WLP-FDTD. As the size of is very small, the conventional WLP-FDTD calculated in this subdomain is barely time consuming. Furthermore, the iteration is not necessary in the proposed method as the splitting error does not exist in the fine structure area. Thus the efficiency has improved.
Figure 4 depicts the error of at the observation point which is defined by , where is the results solved by the efficient WLP-FDTD and the proposed method, respectively. is the solution utilizing the conventional WLP-FDTD. The accuracy of the proposed method is obviously improved compared with the original one.
However, the memory storage required in the proposed method is slightly larger than that of the efficient WLP-FDTD. The conventional WLP-FDTD in whose coefficient matrix is not a tridiagonal matrix and the Schur complement system are needful in the proposed method. So the memory storage will be increased inevitably. In this letter, all calculations have been performed on a Core 4 2.93-GHz machine.
4. Conclusion
In this letter, we devise a new efficient algorithm for the 2D WLP-FDTD method based on domain decomposition scheme. The whole computational domain is decomposed into several subdomains. The conventional WLP-FDTD and the efficient WLP-FDTD methods are introduced to different subdomains. And a joint calculation scheme is applied to the new algorithm. Compared with the efficient WLP-FDTD, the proposed algorithm does not need any iteration. Numerical example has verified that the proposed method can save CPU time and improve the computational accuracy. The research for the 3D case will be presented in the further work.
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
This work was supported in part by the National Science Foundation of China under Grant 51477182.